Along with the scenario in which London is actually taken by the Jacobite army during the winter of 1745-6, this question tends to conjure one of the most popular 'what ifs' of the entire Jacobite era. You will receive differing opinions based on a given historian's understanding of the atmosphere in Scotland and of the morale of the Jacobite soldiers after Culloden, but I will attempt to briefly summarize my own thoughts based upon the archival and historiographical source material I have encountered.
To address the question directly, the Jacobites did not fight on after Culloden because the movement's military apparatus and logistical network simply could not support such an endeavor. The reasons for this are varied, but here are the salient points:
After eight months of mobile campaigning and constant marching, the irregular, non-professional force of around 5500 men who trickled off the moor at Culloden was really in no condition to continue the fight. Between the British army's horrid treatment of wounded enemy soldiers on the battlefield and their chase after the rebel remnants into Inverness, the cohesion of the Jacobite army was torn apart. Despite some calls to continue the fight ahead of the post-battle debriefing (read: disbandment) at Ruthven Barracks in Badenoch, morale was irreparably damaged by shock, confusion, and hunger. The relationships between the commanders had been riven along with it, crescendoing into a web of suspicion and blame that is palpable throughout the few memoirs released in the years after the rising.
Culloden spelled the death of practical Jacobitism, regardless of the roots of its ideologies continuing to live on. In other words, Jacobitism was only effective when expressed militarily as an alternative platform to challenge the Hanoverian regime. To wit, the mix of political, religious, and dynastic ideologies that encompassed Jacobitism and simmered between the 1715 and 1745 risings was largely sustained by a select cadre of elites who had the resources, connections, and time to scheme and toast to the king over the water. A large part of the failure of the Forty-five was due to the inability of Jacobite ideology to permeate deeply into the British populace; on the scale of popular 'rebellions' during the century, the Jacobite effort was quite a small showing – even at its height of popularity around 1715. Because the defeat at Culloden was so complete and so brutal after such a taxing campaign, a potent military force could not have been salvaged from the remains. There were simply not enough willing adherents and no logistical system of support outwith the British Isles. France, as we know, promised plenty but delivered precious little when the time was most ripe. And in a way, the entire rising was predicated on significant and sustained assistance from Louis XV in the form of men, money, and materiel.
Coming back to the fault lines in the Jacobite command exposed by the debacle at Culloden, Lord George Murray – perhaps Charles Edward Stuart's most capable general – resigned his commission at Ruthven and was done with Jacobitism after thirty years of putting his money where his mouth was. The remnants of the army dispersed, encouraged and in fact commanded to do so by the Prince Regent, whereupon his fabled flight from Scotland back to Europe has left us to muse upon 275 years of romantic imagery, fate-filled prose, and political atavism steeped in the modern guise of historical Jacobitism. Ruthven was the proverbial nail in the coffin.
So no cohesive army, few trusted leaders, no logistical base of operations, no widespread popularity, and no method of consistent recruitment. By Culloden the troops were neither getting paid nor fed, and widespread desertion even before the final battle was a serious issue for Jacobite officers, who grew increasingly anxious about maintaining full ranks as the campaign wore on. Government intelligence from the Highlands in 1747 identified significant but scattered rural communities of recalcitrance where Jacobite support remained, but there was no leadership or direction coming from the Stuart court in Europe after the failure at Culloden. Despite the storied cache of gold sent for Jacobite aims allegedly buried at Loch Arkaig, it was but a drop in the bucket of what would have been required to launch yet another shot. Plotting continued between elites after 1746, but the spark of popular adherence was out; it could be rightfully argued that spark never actually caught flame, even with Charles Edward's early successes the year before.
Jacobite sentiment encompassed a number of aims and ideals, especially late into its life. Part of the reason that a protracted guerrilla campaign waged from the Highlands never took shape was because the conflict was not one between Highlanders and Lowlanders. The deepest base of support during the Forty-five came from the largely Episcopalian North-east Lowlands, despite the popular vision of Jacobitism as strictly a Highland affair. Less than half of the clans were supportive of the rising and Jacobitism can much better be described as a British civil war tied to an international context than anything resembling Scotland versus England or Catholics versus Protestants. It was the Highlands, however, which bore the brunt of the ferocious reprisals by British army soldiers in the months after Culloden. With the help of an extensive network of military roads built after 1715, a large-scale cantonment scheme established by the British state, and the watchful activities of loyalist clan chiefs and local sheriffs, growing the seed of another rebellion in the heart of occupied territory – where owning arms had again been made illegal – was simply not feasible.
Finallly, the mainstay of Jacobitism, as espoused by the last viable claimants to the throne of the Three Kingdoms of Britain and Ireland, was never about holding up in the Highlands and conducting clandestine warfare against the British state. The Stuarts were an ancient and established royal dynasty who supremely believed in the Divine Right of kings and their right to rule. They were the British state. Charles Edward was totally convinced that his father was destined to sit upon the throne and oversee a regnal union of all three kingdoms with every British subject being loyal to their family. It was never going to be just Scotland, which is why the retreat from Derby in December 1745 was such a blow to Charles Edward's hopes for a united Stuart Britain. Indeed, it has been considered that the final Jacobite rising ended on that very day rather than 16 April of the following year.
Below are a few published works that carry these points with some greater detail. I hope you find them valuable and that the above answer satisfactorily addresses your query. Please feel free to get in touch if you have further questions.
You might also like to take a look through my doctoral thesis, 'Spines of the Thistle: The Popular Constituency of the Jacobite Rising in 1745-6', which specifically deals with the plebeian participants, and includes themes of recruitment, motivation, and demographics. The thesis is available as a free download in the University of St Andrews digital repository.
With best wishes,
Dr Darren S. Layne
Creator and Curator, The Jacobite Database of 1745
(recently of the Institute for Scottish Historical Research, University of St Andrews)